Question

Consider the Helmholtz coil, in which two identical current loops carrying current ( I ) are placed such that the resulting magnetic field is nearly constant in the region between the loops.

a) Derive the optimal distance of separation d of the loops (Hint: force the relevant second derivative to vanish in the center).

b) Using that d, plot the magnetic field in the region between the two coils, along the z axis.

c)    It is possible to create an electrostatic analog of the Helmholtz coil in which one wishes to make the electric field inside as uniform as possible. This is accomplished by dividing a spherical shell into three segments. The top segment is held at a +V potential, the middle is grounded, and the bottom is at -V. Using the separation of variable solution to Laplace’s equation in spherical geometry, calculate the positions (along z) of the segments (i.e., the latitudes) to maximize the uniformity.

d)    Using that optimum geometry, calculate the electric field inside the spherical shell in .

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Answer 1

"(a) Magnetic field due to a coil al it central axis if given by B = ROI/2 R^2/(R^2 + z)^3/2 so magnetic field due to both coil will be B = mu_0IR^2/2 {1/(R^2 + [d/2 + 2)^3/2 + 1/[R^2 + (d/2 - 2)^3)^3/2